Momentum-space calculation of four-boson recombination

Momentum-space calculation of four-boson recombination A. Deltuva Centro de F´ısica Nuclear da Universidade de Lisboa, P-1649-003 Lisboa, Portugal (Received December 12, 2011)

arXiv:1201.2326v1 [physics.atom-ph] 11 Jan 2012

The system of four identical bosons with large two-boson scattering length is described using momentum-space integral equations for the four-particle transition operators. The creation of Efimov trimers via ultracold four-boson recombination is studied. The universal behavior of the recombination rate is demonstrated. PACS numbers: 34.50.-s, 31.15.ac

I.

INTRODUCTION

In 1970 V. Efimov predicted that with vanishing twoparticle binding an infinite number of weakly bound states with total orbital angular momentum L = 0 may exist in the three-particle system [1, 2]. Such a situation takes place if at least two pairs have infinite scattering length a → ∞ [2]. Away from this limit the number of trimers is finite but, depending on a, may be large [1, 2]. This three-particle Efimov effect has an impact on the behavior of more complex few-particle systems. In particular, for each Efimov trimer the existence of two tetramers was predicted in Refs. [3, 4]. The signatures of these states were observed recently in experiments with ultracold atoms where the formation of the tetramers led to a resonant enhancement of the recombination or relaxation processes [5–7]. These first steps in exploring the four-body Efimov physics experimentally also call for accurate theoretical studies of the four-body systems with large a. Following our recent works devoted to the bosonic atom-trimer [8] and dimer-dimer [9] scattering, here we focus on the four-atom recombination process in the system of four identical bosons. The description is based on the exact four-particle equations for the transition operators [10] derived by Alt, Grassberger, and Sandhas (AGS). For the recombination process we need the transition operator connecting two- and four-cluster channel states; its relation to the two-cluster AGS operators calculated in Refs. [8, 9, 11, 12], will be given in the present work. We solve the integral AGS equations using momentum-space framework. In previous works we demonstrated its reliability for reactions involving many very weakly bound trimers as it happens in the universal regime where the accuracy of the coordinate-space methods may be more limited [4, 13, 14]. For example, in Ref. [15] we predicted universal intersections of shallow tetramers with the corresponding atom-trimer thresholds while the adiabatic hyperspherical calculations of Refs. [4, 6] were not sufficiently precise to find this remarkable feature that leads to a resonant behavior in the ultracold atom-trimer collisions. Thus, we expect that also in the case of the four-boson recombination we will be able to achieve the universal limit with higher accuracy compared to the existing coordinate-space calculations [4, 16]. We note that approximate semi-analytical recombination results have

been obtained in Ref. [17] for a system of three identical bosons plus a distinguishable particle. An extension of the four-body scattering calculations above the four-cluster breakup threshold is a very serious theoretical challenge both in atomic and nuclear physics. In the momentum-space framework the difficulties arise due to complicated singularity structure in the kernel of the integral equations. Not all but some of these difficulties are present already at the four-cluster breakup threshold. In this work we restrict ourselves to a latter case which is sufficient to calculate the four-atom recombination in the ultracold limit. Nevertheless, the present work is an important intermediate step towards solving the momentum-space four-body scattering equations at positive energies. We use a system of units where ~ = 1 and therefore is omitted in the equations unless needed for dimensional analysis. In Sec. II we recall the AGS equations and derive the relation between the two- and four-cluster transition operators. In Sec. III we present results for the four-boson recombination. We summarize in Sec. IV. II.

FOUR-BOSON SCATTERING

We consider a system of four spinless particles with Hamiltonian H = H0 +

6 X

vi ,

(1)

i=1

where H0 is the kinetic energy operator for the relative motion and vi the short-range potential acting within the pair i; there are six pairs. For the description of the scattering process in such a system we use the AGS equations [10]. They are exact quantum-mechanical equations of the Faddeev-Yakubovsky (FY) type. However, in contrast to the original FY equations [18] for the wavefunction components, the AGS equations are formulated for the transition operators in the integral form and therefore are better suited to be solved in the momentumspace framework preferred by us. We aim to determine the amplitude for the recombination of four free particles into a two-cluster state. Due to time reversal symmetry it is equal to the amplitude for the four-cluster breakup of the initial two-cluster state.

2 The latter is more directly related to the AGS transition operators calculated in our previous works [8, 9, 11, 12]. A.

AGS equations

The full wave function corresponding to the initial twocluster channel state |Φρ,n i is also determined by the AGS transition operators, i.e., X ki i |Ψρ,n i = |Φρ,n i + G0 tj G0 Uγjk G0 tk G0 Uγρ |φρ,n i. γjki

In a compact notation the AGS equations can be written as 18-component matrix equations [10, 14]. The components are distinguished by the chains of partitions, i.e., by the two-cluster partition and by the three-cluster partition. Obviously, the two-cluster partitions may be of 3 + 1 or 2 + 2 type; we will denote them by Greek subscripts. All three-cluster partitions are of 2 + 1 + 1 type and are specified by the pair of particles that we will denote by Latin superscripts. For our consideration we need the explicit form of the AGS equations for the transition operators, i.e., X ji ki Uσρ = (G0 ti G0 )−1 δ¯σρ δji + δ¯σγ Uγjk G0 tk G0 Uγρ . (2) γk

Here δ¯σρ = 1 − δσρ , G0 = (E + i0 − H0 )−1

(3)

is the free resolvent with the available four-particle energy E, ti = vi + vi G0 ti

(4)

is the two-particle transition matrix for the pair i, and X ¯ δ¯ji ti G0 Uγik (5) Uγjk = G−1 0 δjk + i

is the subsystem transition operator of 3+1 or 2+2 type, depending on γ. In the summations over the pairs only the ones internal to the respective two-cluster partition contribute. ji The transition operators Uσρ contain full information on the scattering process. Their on-shell matrix elements P j ji i ji hφσ,n′ |Uσρ |φρ,n i are the amplitudes hΦσ,n′ |Tσρ |Φρ,n i for the two-cluster reactions. Here X δ¯ij tj |φjρ,n i (6) |φiρ,n i = G0 j

are the Faddeev components of the initial and final channel states X |φiρ,n i, (7) |Φρ,n i = i

and n distinguishes between the different channel states in the same partition. |Φρ,n i is given by the nth bound state wave function in partition ρ times the plane wave with momentum pρ,n between the clusters. and is normalized to hΦσ,n′ |Φρ,n i = δσρ δn′ n δ(pσ,n′ − pρ,n ). The on-shell condition for each channel state relates its binding energy, relative two-cluster momentum and reduced mass as ǫρ,n + p2ρ,n /2µρ = E.

(8) The amplitude for the four-cluster breakup reaction can be obtained as hΦ0 |T0ρ |Φρ,n i = hΦ0 |

6 X i=1

vi |Ψρ,n i.

(9)

The wave function (8) satisfies also the Schr¨odinger equaP6 −1 tion, thus, i=1 vi |Ψρ,n i = G0 |Ψρ,n i. Furthermore, −1 G0 |Φ0 i = 0 since the four-cluster channel state |Φ0 i is an eigenstate of H0 with eigenvalue E. In fact, |Φ0 i is a product of three plane waves (each is normalized to the Dirac δ-function) corresponding to the relative motion of four free particles. Thus, the amplitude for the four-cluster breakup of the two-cluster initial state is X ki i hΦ0 |T0ρ |Φρ,n i = hΦ0 |tj G0 Uγjk G0 tk G0 Uγρ |φρ,n i. γjki

(10) Due to time reversal symmetry it describes also the four-particle recombination into a two-cluster state, i.e., hΦρ,n |Tρ0 |Φ0 i = hΦ0 |T0ρ |Φρ,n i. If all four particles are identical, there are only two distinct two-cluster partitions, one of 3+1 type and one of 2 + 2 type. We choose those partitions to be ((12)3)4 and (12)(34) and denote them in the following by α = 1 and 2, respectively. The AGS equations (2) for the symmetrized transition operators Uβα become U11 = P34 (G0 tG0 )−1 + P34 U1 G0 tG0 U11 + U2 G0 tG0 U21 , (11a) −1

U21 = (1 + P34 )(G0 tG0 )

+ (1 + P34 )U1 G0 tG0 U11 , (11b)

U12 = (G0 tG0 )−1 + P34 U1 G0 tG0 U12 + U2 G0 tG0 U22 , (11c) U22 = (1 + P34 )U1 G0 tG0 U12 .

(11d)

Here the two-particle transition matrix t acts within the pair (12) and the symmetrized operators for the 3+1 and 2+2 subsystems are obtained from the integral equations Uα = Pα G−1 0 + Pα tG0 Uα .

(12)

The employed basis states have to be symmetric under exchange of two particles in subsystem (12) for 3 + 1 partition and in (12) and (34) for 2 + 2 partition. The correct symmetry of the four-boson system is ensured by the operators P34 , P1 = P12 P23 + P13 P23 , and P2 = P13 P24 where Pab is the permutation operator of particles a and b. For each two-cluster channel state |Φα,n i =

3 (1 + Pα )|φα,n i there is only one independent Faddeev amplitude obtained from the integral equation |φα,n i = G0 tPα |φα,n i.

(13)

The symmetrized amplitudes for the two-cluster reactions are given in Refs. [8, 9, 11, 12]. The four-cluster breakup amplitude in terms of the symmetrized AGS operators (11) becomes hΦ0 |T0α |Φα,n i = S0α hΦ0 |(1 + P1 ){[1 + P34 (1 + P1 )] × t G0 U1 G0 t G0 U1α + (1 + P2 )t G0 U2 G0 t G0 U2α }|φα,n i. (14) The factors S0α depend on the symmetry √ of the final channel state |Φ0 i. For example, S01 =√ 6 and S02 = √2 for nonsymmetrized |Φ0 i, while S01 = 3 and S02 = 2 for |Φ0 i that is symmetrized within the pair (12). We denote by |Φ00 i the four-cluster channel state at threshold, where E = 0 and all momenta vanish. Since all particles have equal momenta, the channel state |Φ00 i is invariant under all permutations, i.e., Pab |Φ00 i = |Φ00 i. In this particular case the relation (14) simplifies to hΦ00 |T0α |Φα,n i = S0α hΦ00 |(12 t G0 U1 G0 t G0 U1α + 6 t G0 U2 G0 t G0 U2α )|φα,n i.

(15)

We use the momentum-space partial-wave framework to solve the AGS equations (11). Two different types of basis states |kx ky kz [(lx ly )Jlz ]J Miα = |kx ky kz νiα with α = 1 and 2 are employed. Here kx , ky , and kz denote magnitudes of the Jacobi momenta. For α = 1 the Jacobi momenta describe the relative motion in the 1+1, 2+1, and 3+1 subsystems and are expressed in terms of single particle momenta ka as 1 (k2 − k1 ), 2 1 ky = [2k3 − (k1 + k2 )], 3 1 kz = [3k4 − (k1 + k2 + k3 )], 4 while for α = 2 they describe the relative motion 1+1, 1+1, and 2+2 subsystems, i.e., kx =

(16a) (16b) (16c) in the

1 (k2 − k1 ), (17a) 2 1 ky = (k4 − k3 ), (17b) 2 1 kz = [(k4 + k3 ) − (k1 + k2 )]. (17c) 2 The respective orbital angular momenta lx , ly , and lz are coupled to the total angular momentum J with the projection M; all discrete quantum numbers are abbreviated by ν. An explicit form of integral equations is obtained by inserting the respective completeness relations XZ ∞ 1= |kx ky kz νiα kx2 dkx ky2 dky kz2 dkz α hkx ky kz ν| kx =

ν

between all operators in Eqs. (11). Due to rotational symmetry all operators are diagonal in J and independent of M. In addition, Uα are diagonal in kz , J and lz , P34 is diagonal in kx and lx , t is diagonal in ky , kz and all ν, and G0 is diagonal in all quantum numbers. In this representation the AGS equations for each J become a system of coupled integral equations in three continuous variables kx , ky , and kz . Such threevariable equations have been solved in Refs. [11, 12] for the four-nucleon scattering below the three-cluster breakup threshold. The kernel of integral equations (11) contains integrable singularities arising from each subsystem bound state pole of Uα ; we isolate these poles in different subintervals and treat them by the subtraction technique when integrating over kz [11]. However, in the energy regime of the four-cluster breakup, E ≥ 0, the AGS equations (11) contain additional singularities arising from G0 , namely, α hkx′ ky′ kz′ ν ′ |G0 |kx ky kz νiα = δν ′ ν δ(kx′ − kx )δ(ky′ − ky )δ(kz′ − kz )/[kx2 ky2 kz2 (E + i0 − kx2 /2µαx − ky2 /2µαy − kz2 /2µα )], where µαx and µαy are the respective reduced masses. The operator products like P34 P1 G0 render the singularity structure of Eqs. (11) very complicated. It is considerably simpler at E = 0 where there is only one singular point kx = ky = kz = 0. In this case the integrals involving G0 over any of the Jacobi momenta kj (except for the trivial integrals involving momentum δ-functions) are of type Z

0

∞

f (kx , ky , kz )kj2 dkj i0 − kx2 /2µαx − ky2 /2µαy − kz2 /2µα

(19)

where the function f (kx , ky , kz ) is regular at kj = 0. Thus, the the singularity of G0 is cancelled by kj2 making the integral numerically harmless. Nevertheless, G0 varies very rapidly near kx = ky = kz = 0 and therefore one should avoid interpolation of G0 ; this will be explained in Sec. II B. The four free boson channel state |Φ00 i has kx = ky = kz = 0 and lx = ly = lz = J = J = 0. Due to the threshold law the only nonvanishing partial-wave component of the breakup amplitude (15) is the one with J = 0, i.e., J =0 |Φα,n i/(4π)2 . hΦ00 |T0α |Φα,n i = hΦ00 |T0α

(20)

Although |Φ00 i has only the lj = J = J = 0 component, we emphasize that angular momenta lj and J are not conserved. Nonzero lj and J basis states must be included when solving the AGS equations (11).

B.

Separable potential

The universal properties of the four-boson system must be independent of the short-range interaction details. We choose the two-boson potential of a separable form acting in the S-wave only, i.e.,

0

(18)

v = |giλhg| δlx 0 .

(21)

4 The form factor hkx |gi = [1 + c2 (kx /Λ)2 ]e−(kx /Λ)

2

(22)

is taken over from Ref. [8]. Two very different choices of c2 , namely, c2 = 0 and c2 = −9.17, are used in order to confirm the universality of the obtained results. The strength −1     2 3c2 1 Λ c2 √ (23) 1+ λ= − 1+ πm a 2 8 2π is constrained to reproduce the given value of the scattering length a for two particles of mass m. The potential (21) supports one shallow two-boson bound state at a > 0 and no one at a < 0. There are no deeply bound dimers. The resulting two-boson transition matrix t = |giτ hg|

(24)

is separable as well with τ = (λ−1 − hg|G0 |gi)−1 . This allows to reduce the AGS equations (11) to a system of two-variable (ky and kz ) integral equations and thereby simplifies considerably the numerical calculations. The AGS equations with separable interactions are solved in the form hg|G0 U11 |φ1,n i = P34 hg|P1 |φ1,n i + P34 hg|G0 U1 G0 |giτ hg|G0 U11 |φ1,n i + hg|G0 U2 G0 |giτ hg|G0 U21 |φ1,n i, (25a)

The discretization of integrals in Eqs. (25) using Gaussian quadrature rules leads to a system of linear algebraic equations. We reduce its dimension even further by applying the so-called K-matrix technique [14] converting the complex equations into the real ones, and by eliminating the α = 2 component, i.e., by inserting Eq. (25b) into (25a) and Eq. (25d) into (25c). To avoid difficulties due to possibly slow convergence of iterative methods, the resulting linear system is solved using the direct matrix inversion. Further details of the calculations can be found in Refs. [11, 14].

III.

FOUR-BOSON RECOMBINATION

The number of the four-atom recombination events in a non-degenerated atomic gas per volume and time is K4 ρ4 where ρ is the density of atoms and K4 the fouratom recombination rate. In the ultracold limit the kinetic energies of initial atoms are much smaller than the final two-cluster kinetic or binding energies, and the initial momenta are much smaller than the momenta of the two resulting clusters, kx , ky , kz 1 not only the ground state but also excited trimers are produced via four-atom recombination

6 0

10

-1

10

n=4

n’ = 0 n’ = 1 n’ = 2 n’ = 3

νn,n’

-2

10

-3

10

-4

10

-5

10

0.2

0.4

0.6 a/a0n

0.8

1.0

FIG. 3. (Color online) Relative weights νn,n′ for the production of the n′ th state trimers via four-boson recombination. n = 4 atom-trimer channels contribute in the shown regime.

and contribute to the total rate (26). In Fig. 3 we compare the relative weights νn,n′ for the production of the trimers in the individual states n′ when n atom-trimer channelsP are available in total. The weights are normaln−1 ized to n′ =0 νn,n′ = 1. The results in Fig. 3 clearly demonstrate that more weakly bound trimers are produced more efficiently and the four-atom recombination process is strongly dominated by the n′ = (n − 1)th channel with the shallowest available trimer. We note that a similar conclusion was drawn in Ref. [9] about the trimer production in dimer-dimer collisions. Except for the vicinity of a/a0n = e−π/s0 where the shallowest trimer disappears, the relative weights νn,n′ depend quite weakly on a. Thus, the tetramer states around a = a0n,k enhance the production of all trimers by equal factors. As mentioned in the introduction, the recombination of four identical bosons has been calculated in Refs. [4, 16] using the adiabatic hyperspherical framework. There is a good qualitative agreement between our results and those of Refs. [4, 16]: the shape of the recombination rate, including its resonant peaks and cusps, is reproduced in both cases. However, the calculations of Refs. [4, 16] are limited to n = 1 where the finite-range effects are not entirely negligible and deviations from our universal results can be expected much like in the case of our n = 1 results in Fig. 2. Within this accuracy the predictions of

[1] V. Efimov, Phys. Lett. B 33, 563 (1970). [2] E. Braaten and H.-W. Hammer, Phys. Rep. 428, 259 (2006). [3] H. W. Hammer and L. Platter, Eur. Phys. J. A 32, 113 (2007). [4] J. von Stecher, J. P. D’Incao, and C. H. Greene, Nature Phys. 5, 417 (2009). [5] F. Ferlaino, S. Knoop, M. Mark, M. Berninger, H. Sch¨ obel, H.-C. N¨ agerl, and R. Grimm, Phys. Rev. Lett.

Refs. [4, 16] for the peak positions, i.e., a0n,1 /a0n ≈ 0.43 and a0n,2 /a0n ≈ 0.90, are consistent with our results. On the other hand, the available experimental data [6] are obtained in the system of ultracold 133 Cs atoms with deeply bound non-Efimov-like few-atom states and refer to n = 0. Therefore it is not surprising that the experimental results a0n,1 /a0n ≈ 0.47 and a0n,2 /a0n ≈ 0.84 deviate from the universal values even more. Nevertheless, they remain roughly consistent with the theoretical predictions. Thus, our calculations present no improvement over the ones of Refs. [4, 16] when comparing with the existing data, but serve as a theoretical benchmark and may be valuable for the future experiments performed in a truly universal regime. IV.

SUMMARY

We studied ultracold four-boson recombination. Exact four-particle scattering equations in the AGS version have been used. We related the four-cluster breakup and recombination operators to the standard two-cluster AGS operators. The scattering equations at the fourcluster breakup threshold have been precisely solved in the momentum-space framework. The zero-temperature limit of the four-boson recombination rate was calculated for negative values of the two-boson scattering length. We demonstrated its universal behavior in the regime with high excited Efimov trimers where the finite-range effects are negligible, and determined the positions of resonant recombination peaks caused by the tetramer states. We also have shown that most of the Efimov trimers produced via the four-boson recombination are in the state with weakest binding. Our results are consistent with existing experimental data [6] and theoretical predictions [4, 16], but exceed the latter ones with respect to the accuracy in the universal limit. In addition, the present work constitutes a step towards solving the momentumspace four-body scattering equations above the breakup threshold. ACKNOWLEDGMENTS

The author thanks P. U. Sauer for discussions. on fewbody recombination problem.

101, 023201 (2008). [6] F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P. D’Incao, H.-C. N¨ agerl, and R. Grimm, Phys. Rev. Lett. 102, 140401 (2009). [7] S. E. Pollack, D. Dries, and R. G. Hulet, Science 326, 1683 (2009). [8] A. Deltuva, Phys. Rev. A 82, 040701(R) (2010). [9] A. Deltuva, Phys. Rev. A 84, 022703 (2011).

7 [10] P. Grassberger and W. Sandhas, Nucl. Phys. B2, 181 (1967); E. O. Alt, P. Grassberger, and W. Sandhas, JINR report No. E4-6688 (1972). [11] A. Deltuva and A. C. Fonseca, Phys. Rev. C 75, 014005 (2007). [12] A. Deltuva and A. C. Fonseca, Phys. Rev. C 76, 021001(R) (2007). [13] R. Lazauskas and J. Carbonell, Phys. Rev. A 73, 062717 (2006). [14] A. Deltuva, R. Lazauskas, and L. Platter, Few-Body Syst. 51, 235 (2011).

[15] A. Deltuva, EPL 95, 43002 (2011). [16] N. P. Mehta, S. T. Rittenhouse, J. P. D’Incao, J. von Stecher, and C. H. Greene, Phys. Rev. Lett. 103, 153201 (2009). [17] Y. Wang and B. D. Esry, Phys. Rev. Lett. 102, 133201 (2009). [18] O. A. Yakubovsky, Yad. Fiz. 5, 1312 (1967) [Sov. J. Nucl. Phys. 5, 937 (1967)].